Quantum systems are typically ergodic and can be described by a thermal ensemble. In this work, the authors use large-scale tensor network numerics to investigate if a generic two-dimensional disordered Heisenberg model of quantum spins exhibits a nonergodic phase. They provide evidence for the existence of a many-body localized regime at sufficiently strong disorder. They also estimate the critical disorder strength where a crossover into an ergodic regime occurs and address finite-size effects.
Quantum systems are typically ergodic and can be described by a thermal ensemble. In this work, the authors use large-scale tensor network numerics to investigate if a generic two-dimensional disordered Heisenberg model of quantum spins exhibits a nonergodic phase. They provide evidence for the existence of a many-body localized regime at sufficiently strong disorder. They also estimate the critical disorder strength where a crossover into an ergodic regime occurs and address finite-size effects.
Previous studies reveal a crucial effect of symmetries on the properties of a single particle moving in a disorder potential.More recently, a phenomenon of many-body localization (MBL) has been attracting much theoretical and experimental interest.MBL systems are characterized by the emergence of quasilocal integrals of motion and by the area-law entanglement entropy scaling of its eigenstates.In this paper, we investigate the effect of a non-Abelian SUð2Þ symmetry on the dynamical properties of a disordered Heisenberg chain.While SUð2Þ symmetry is inconsistent with conventional MBL, a new nonergodic regime is possible.In this regime, the eigenstates exhibit faster than area-law, but still strongly subthermal, scaling of the entanglement entropy.Using extensive exact diagonalization simulations, we establish that this nonergodic regime is indeed realized in the strongly disordered Heisenberg chains.We use the realspace renormalization group (RSRG) to construct approximate excited eigenstates by tree tensor networks and demonstrate the accuracy of this procedure for systems of sizes up to L ¼ 26.As the effective disorder strength is decreased, a crossover to the thermalizing phase occurs.To establish the ultimate fate of the nonergodic regime in the thermodynamic limit, we develop a novel approach for describing many-body processes that are usually neglected by the RSRG.This approach is capable of describing systems of size L ≈ 2000.We characterize the resonances that arise due to such processes, finding that they involve an ever-growing number of spins as the system size is increased.Crucially, the probability of finding resonances grows with the system's size.Even at strong disorder, we can identify a large length scale beyond which resonances proliferate.Presumably, this proliferation would eventually drive the system to a thermalizing phase.However, the extremely long thermalization timescales indicate that a broad nonergodic regime will be observable experimentally.Our study demonstrates that, similar to the case of single-particle localization, symmetries control dynamical properties of disordered, many-body systems.The approach introduced here provides a versatile tool for describing a broad range of disordered manybody systems, well beyond sizes accessible in previous studies.
When pushed out of equilibrium, generic interacting quantum systems equilibrate locally and are expected to evolve towards a locally thermal description despite their unitary time evolution. Systems in which disorder competes with interactions and transport can violate this expectation by exhibiting many-body localization. The strength of the disorder with respect to the other two parameters drives a transition from a thermalizing system towards a nonthermalizing one. The existence of this transition is well established both in experimental and numerical studies for finite systems. However, the stability of many-body localization in the thermodynamic limit is largely unclear. With increasing system size, a generic disordered system will contain with high probability areas of low disorder variation. If large and frequent enough, those areas constitute ergodic grains which can hybridize and thus compete with localization. While the details of this process are not yet settled, it is conceivable that if such regions appear sufficiently often, they might be powerful enough to restore thermalization. We set out to shed light on this problem by constructing potential landscapes with low disorder regions and numerically investigating their localization behavior in the Heisenberg model. Our findings suggest that many-body localization may be more stable than anticipated in other recent theoretical works.
In recent years, physicists have made great strides toward better understanding isolated many-body quantum systems. Already, two distinct phases are well known: systems that are ergodic and thermalize completely, and, by contrast, systems that exhibit many-body localization (MBL) due to a strong disorder potential, which prevents thermalization. It is tempting to wonder whether a phase of matter could exist between the extremes of full thermalization and MBL in a generic, i.e., nonintegrable, isolated many-body quantum system. Here, the authors provide evidence that such a nonergodic phase can be realized in a translationally invariant itinerant electron model, with both charge and spin degrees of freedom. Exact diagonalization calculations on finite system sizes reveal that there exists a ``spin band'' of nonthermal eigenstates. These states have entanglement entropy that scales as an overall ``volume law,'' but after a partial measurement of the spin on each site the scaling is modified to an ``area law,'' thus suggesting the states are not ergodic. These properties and others suggest that this model realizes a new nonthermal phase of matter, known as a quantum disentangled liquid (QDL). The putative existence of this phase has striking implications for the foundations of quantum statistical mechanics.
Many-body localization (MBL) enables new nonergodic dynamical phases of matter that avoid the fate of thermalization. Do other ergodicity-breaking phases beyond the MBL paradigm exist? Here, the authors introduce a method for constructing and probing the stability of candidate nonergodic phases. As an example, they consider disordered spin systems, in which a non-Abelian SU(2) symmetry precludes the conventional MBL phase that is characterized by quasilocal integrals of motion and area-law entanglement entropy of eigenstates. Instead, the authors construct a candidate nonergodic phase characterized by eigenstates having logarithmic scaling of entanglement entropy. They find that the phase is only marginally unstable and, therefore, nonthermal yet non-MBL dynamics can be observed in such disordered systems.
Understanding the effects of disorder in quantum many-body systems is one of the central challenges of contemporary statistical physics. Its interplay with interactions and quantum fluctuations profoundly modifies the nature of quantum states and their dynamics, giving rise to phenomena such as localization, ergodicity breaking, and the emergence of complex collective behavior. This thesis presents new results on two classes of disordered quantum systems. The first part is devoted to the study of the many-body localization transition, a paradigmatic example of disorder-induced ergodicity breaking. The critical properties of the transition are investigated using a numerical renormalization-group approach, establishing a connection between its scaling behavior and that of the Anderson localization transition on tree-like graphs. The second part focuses on the zero-temperature quantum spin-glass phase of the two-dimensional bond-disordered Heisenberg model. Combining large-scale numerical simulations with a semiclassical expansion, evidence is provided for the existence of the spin-glass phase. The effective model governing the semiclassical fluctuations is further analyzed to characterize the low-energy excitation spectrum and its localization properties. Finally, the weak-disorder regime is investigated and its implications for the ferromagnetic phase are discussed. Overall, the results presented in this thesis contribute to the understanding of the effects of disorder in interacting quantum many-body systems, encompassing both localization phenomena and the collective behavior of disordered quantum magnets.
We study high temperature spin transport in a disordered Heisenberg chain in the ergodic regime. By employing a density matrix renormalization group technique for the study of the stationary states of the boundary-driven Lindblad equation we are able to study extremely large systems (400 spins). We find both a diffusive and a subdiffusive phase depending on the strength of the disorder and on the anisotropy parameter of the Heisenberg chain. Studying finite-size effects, we show numerically and theoretically that a very large crossover length exists that controls the passage of a clean-system dominated dynamics to one observed in the thermodynamic limit. Such a large length scale, being larger than the sizes studied before, explains previous conflicting results. We also predict spatial profiles of magnetization in steady states of generic nondiffusive systems.